# The Excel IMAGINARY Function

Complex Numbers

Within the world of complex numbers, the 'imaginary' value, i is used to represent the square root of -1.

A Complex Number is composed of a real number combined with an imaginary number.

E.g. The complex number, z, is written as

z   =   5 + 2i
Related Function:
IMREAL Function

## Function Description

The Excel IMAGINARY function returns the imaginary coefficient of a supplied complex number.

I.e. for the complex number a + bi, the Excel Imaginary function returns the value b.

The syntax of the function is:

IMAGINARY( inumber )

where the inumber argument is a complex number.

### Complex Numbers in Excel

Note that complex numbers are simply stored as text in Excel. When a text string in the format "a+bi" or "a+bj" is supplied to one of Excel's built-in complex number functions, this is interpreted as a complex number.

Also the complex number functions can accept a simple numeric value, as this is equivalent to a complex number whose imaginary coefficient is equal to 0.

Therefore, the inumber argument can be supplied to the Excel Imaginary function as any of the following:

• A simple number;
• A complex number encased in quotation marks - e.g. "5+3i";
• A reference to a cell containing a complex number or a numeric value;
• A value returned from another Excel function or formula.

## Excel Imaginary Function Examples

In the example spreadsheet below, the Excel Imaginary function is used find the imaginary coefficient of five different complex numbers.

Formulas:
AB
15+2i=IMAGINARY( A1 )
2 =IMAGINARY( "2-i" )
3 =IMAGINARY( 6 )
4 =IMAGINARY( "3i" )
5 =IMAGINARY( COMPLEX( 4, 1 ) )
Results:
AB
15+2i2
2 -1
3 0
4 3
5 1

Note that, in the above example spreadsheet:

• The integer 6, used in cell B3, is equal to the complex number 6+0i;
• The example in cell B5 uses the Excel Complex Function to create the complex number 4+i.

Further details of the Excel Imaginary function are provided on the Microsoft Office website.

## Imaginary Function Errors

If you get an error from the Excel Imaginary Function, this is likely to be one of the following:

Common Errors
 #NUM! - Occurs if the supplied inumber argument is not recognised as a complex number. #VALUE! - Occurs if the supplied inumber argument is a logical value.